How to generate the following Dataset to plot in 3D

Question

I have been trying for around 2 days now to create the 2 half-moons data set in 3 dimensions.

First, I did it in two dimensions, and this link is very helpful for understanding how to plot the data set: Creating a dataset in the form of two half semi circle arcs in python

Then, I decided to try 3 dimensions. The way to do this would be to think of the shape as a tube that is bent, and this tube is composed of a lot of thin disks. In each of these disks, we take a uniform random number of points, which this link is very useful for: Generate a random point within a circle (uniformly)

After this, I need to rotate and translate these disks, but this is what I am stuck on and would appreciate any help. If you are unable to help and provide code or explanation for this, another approach could also be very helpful.

Answer 1

Then, I decided to try 3 dimensions. The way to do this would be to think of the shape as a tube that is bent

Seems like you are trying to generate random points inside the volume of a torus (or half a torus). Instead of following your approach (a series of disks), I suggest a different one. First, you need to parameterize the torus.

Then, you need to generate the random distribution of points.

Let's put the pieces together. I'm going to use Plotly as it has an equal aspect ratio for 3d plots. Obviously, you can also apply offsets to the dataset.

import numpy as np

def random_dist_torus(a, c, n_low=1000, n_high=10000):
    """
    Parameters
    c: radius from the center of the hole to the center of the torus
    a: radius of the tube
    n_low: minimum number of points in the volume
    n_high: maximum number of points in the volume
    """
    n = int(np.random.uniform(n_low, n_high))
    # consider a cross section of the torus, which is going
    # to be a circle. The variable r represents the radius
    # of the points in this cross section.
    r = a * np.random.uniform(0, 1, n)
    # u: angular coordinate of the torus. Here I set it
    # to np.pi, so it will generate only half torus
    u = np.pi * np.random.uniform(0, 1, n)
    # v: circumferential coordinate. Consider a cross section
    # of a torus, which is going to be a circle. v represents
    # the angle around this circle.
    v = 2 * np.pi * np.random.uniform(0, 1, n)
    return (
        (c + r * np.cos(v)) * np.cos(u), # x-coordinates
        (c + r * np.cos(v)) * np.sin(u), # y-coordinates
        r * np.sin(v)  # z-coordinates
    )


import plotly.graph_objects as go

x1, y1, z1 = random_dist_torus(2, 4)
x2, y2, z2 = random_dist_torus(2, 4)
fig = go.Figure(
    data=[
        go.Scatter3d(
            x=x1, y=y1, z=z1,
            mode='markers',
            marker=dict(
                size=2,
            )
        ),
        go.Scatter3d(
            x=x2, y=-y2, z=z2,
            mode='markers',
            marker=dict(
                size=2,
            )
        )
    ])
fig.show()